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Introduction

Mathematics Coursework: T-Totals

T-Totals – 10x10 Grid  1+2+3+12+22= 40

T-Number=22                                                        T-Total= 40 24+25+26+35+45= 145

T-Number=45

T-Total=145 61+62+63+72+82= 340

T-Number=82

T-Total=340 78+79+80+89+99= 425

T-Number=99

T-Total=425

For 10x10 Grid:

• When the T-number is even, so is the T-Total.
• When the T-number is odd, so is the T-Total.

I will now find the rule which links the T-Number with the T-Total:  n+(n-10)+(n-20)+

(n-21)+(n-19)

=5n-70

When n= 59 =(5x59)-70=225

Testing: 38+39+40+49+59=225

As you can see my rule has worked.

9x9 Grid:  1+2+3+11+20= 37

T-Number=20

T-Total= 37 22+23+24+32+41= 142

T-Number=41

T-Total=142

46+47+48+56+65= 262 T-Number=65

T-Total=262 61+62+63+71+80=337

T-Number=80

T-Total=337

For 9x9 Grid: • When the T-Number is even, the T-Total is odd.
• When the T-Number is odd, the T-Total is even.

Middle

157

8x8 Grid  1+2+3+10+18= 34

T-Number=18

T-Total=34 20+21+22+29+37= 129

T-Number=37

T-Total=129

33+34+35+42+50= 194 T-Number= 50

T-Total= 194 45+46+47+54+62= 254

T-Number= 62

T-Total= 254

For 8x8 Grid:

• When the T-Number is odd, the T-Total is odd.
• When the T-Number is even, the T-Total is even.

I will now find a rule which links the T-number with the T-Total:  n+(n-8)+(n-16)+

(n-18)+(n-17)

=5n-56

When n=36 =(5x36)-56=124

Testing: 19+20+21+28+36=124

As you can see my rule has worked.

T-Totals – Any sized Grid

I will now find the general rule for any sized grid, which links the T-Number with the T-Total.  n+(n-G)+(n-2G)

+(n-2G-1)+(n-2G+1)

= 5n-7G

When n=65, and G=10         =(5x65)-(7x10)= 255 44+45+46+55+65= 225

As you can see my rule has worked.

Translation:

If I translate the T 3 Vectors right, it will become:  22+23+24+33+43= 145                     25+26+27+36+46= 160

T-Number=43                                     T-Number=46

T-Total= 145                                     T-Total=160

• The T-Total has increased by 15.
• This because there 5 numbers in the T-Total, which all have increased by 3
• 5x3=15
• Consequently, if the vector is ( ) the formula for moving T across would be: T-Total+5A

If I translate T, 3 vectors up, it will become:  62+63+64+73+83= 345                  32+33+34+43+53= 195

Conclusion

= Formula for new T-number

5n-7is the formula for T rotated at 270* on point n.

I will now substitute the old ‘n’ (T-number) with the new ‘n’ (the new T-number).

5(n+c-dG+d+cG) -75n+5c-5dG+5d+5cG-7

This is the new formula for the T-Total of my newly rotated shape.

Combination (Rotation & Translation)

• I will now find the general rule for rotating and then translating a T.
• The rule will only work in certain Grids, because the vector by which I want to translate the T, will be too big to fit the limitations of the Grid.
• The limitations are only a problem, if you want the T-shape to stay in the grid, but I have demonstrated already using the 6x6 grid that you can imagine the numbers carrying on.
• To do this I must combine the rules for rotation with the ones for translations:

90*=5n+5c-5dG-5d-5cG+7+5a-5bG

This allows you to rotate the T by 90* and then move it by any vector.

180*=5n+10c-10dG+7G+5a-5bG

This allows you to rotate the T by 180* and then move it by any vector.

270*= 5n+5c-5dG+5d+5cG-7+5a-5bG

This allows you to rotate the T by 270* and then move it by any vector.

This student written piece of work is one of many that can be found in our GCSE T-Total section.

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This is an incredibly well structured and demonstrated algebraic investigation. The staging of the investigation allows for the concepts to be developed throughout. Specific strengths and improvements are suggested throughout. All the mathematical content in the investigation is correct.

Marked by teacher Cornelia Bruce 18/07/2013

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